CH696749A5 - Data processing system for performing risk analysis of portfolio, simulates realization of risk factors by using calibrated correlation matrix, calibration values of parameters, risk mapping function and portfolio data - Google Patents

Abstract

A simulation unit simulates realization of risk factors by using the calibrated correlation matrix, calibration values of the parameters, risk mapping function and the portfolio data. An output unit outputs the simulation data in the form of risk measure or a price. An independent claim is also included for data processing method.

Description

CH 696 749 A5

Description Background of the Invention

The present invention relates to a system for carrying out a risk analysis of a portfolio.

The financial services industry, in particular the departments for the management of financial risks and for the calculation of financial security of insurance and reinsurance companies and banks, has in the past created tools and means for the estimation of their financial risks. Such risks may be related to debt instruments and portfolios of debt instruments such as securities or bonds. Such risks may also be related to equity portfolios of different currencies or to insurance and reinsurance obligations.

These tools and means are based on models on which simulations are performed to generate possible evaluation scenarios. In general, these simulations use the Monte Carlo method or other suitable methods. The models use probability distributions and are matched against historical data. Such historical data can be obtained from a variety of sources, such as DataStream ™.

These simulations are usually implemented by computer programs as part of a financial services system and are executed on computer hardware.

The input data for the simulations are risk factors, which are treated as random variables. Such risk factors may be equity indices, foreign exchange rates, interest rates, or loss of insurance frequencies and degrees. The result or output data of such simulations is at least one measure of risk in the form of a numerical value or a numerical value. Usually, multiple values of risk masses of different types can be obtained.

These values of risk weights are passed to an analyst or actuary or underwriter, i. to a human representative of a financial services company. These values of risk masses allow him to decide if action should be taken to reduce the risk. Such actions may be changes in a credit or equity portfolio or in a portfolio of insurance and reinsurance performance obligations.

The risk mass usually consists of a variety of values, such as the maximum value obtained, the standard deviation of the simulation, a shortfall, usually the 99% shortfall, or a value-at-risk (VAR ™). The VAR ™ is the largest possible loss that the Company can expect with a given given degree of probability for a given future time period in the relevant portfolio. The full distribution itself can also be the risk measure.

Typically, a large number of risk factors must be considered. Therefore, multidimensional probability distributions must be used. Since the risk mass is often determined in the tail of such distributions, accurate modeling of tail dependency is important.

Moreover, the dependence of the risk factors must be taken into account. However, dependency is often not adequately modeled using a linear correlation. A known solution for better model dependency is the use of copulas.

These copulas are well known in the art. They are linked distribution functions of random variables with standard uniform marginal distributions. They provide a way to understand how edge distributions of individual risk are coupled together to form linked distributions of risk groups.

Various types of copulas are known. Examples of closed-type copulas are the Gumpels and the Clayton Copulas. Examples of implicit copulas, i. Copulas for which no closed form exists are Gaussian copula and the t-copula.

It has become increasingly popular to include vectors of risk factor log returns with so-called meta-t distributions, i. Distributions with a t-copula and arbitrary edge distribution, to model. The reason for this is the ability of the t-copula to model the extreme dependence of risk factors, as well as the ease with which the parameters of the t-copula can be estimated from data. The use of such t-copulas for the modeling of loan portfolio losses is described in Frey Rüdiger et al., "Copulas and Credit Models" RISK, October 2001, p.p. 111-114. The related disclosure is hereby incorporated by reference.

Therefore, here we will only reproduce the basic definitions and properties of t-distributions and t-copulas. For more information on copulas in general, see Nelsen, R. (1999): An Introduction to copulas. Springer, New York, or Embrechts, P., A. McNeil, and D. Straumann (2002): "Correlation and Dependence in Risk Management: Properties and Pitfalls," in Risk Management: Value at Risk and Beyond, ed. By M Dempster, pp. 176-223. Cambridge University Press, Cambridge.

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Before describing the prior art and the present invention in more detail, it is helpful to define the various variables and values. The following notation is used in the description of the prior art and the invention:

d dimension, number of risk factors of d-dimensional conventional real vector space

E (X) Expected value of the random variable X

Var (X) Variance of the random variable X

Cov (X, Y) covariance of the random variables XundY

X, Y, Z random vectors

Cov (X) covariance matrix of X v number of degrees of freedom

S covariance matrix

Nd (0, S) d-dimensional Gaussian distribution with

Average value 0 and covariance S <|> univariate Gaussian distribution function

Xv2 Chi Square distribution with degree of freedom v p correlation matrix tv distribution function of student t with

Degree of freedom v tv ~ 1 student's quantitive function t tViPd d-dimensional distribution function of student t with correlation matrix p and degree of freedom v F conventional gamma function det A determinant of matrix A

Ht freely selectable univariate distribution function

U random variable with uniform distribution in

[0.13

t (X, Y) Kendall's tau rank correlation for

Random variables X and Y ak Credit Multifactor Model parameter pUJ Probability of occurrence of the event

A

p [jt probability that X is less than or equal to x

Ä * announcement business idiosyncratic parameter

Ek credit risk on announcement business lk loss given default

Let z ~ Nd (0,2) and U (uniformly distributed random variable in [0,1]) be independent Moreover, G denotes the distribution function of V ^ î and R = G "1 (U).

Then, the Rd - scored random vector Y, which is given by

Y = (A Zi, R 2i, R Z3, -, R Za) '(!)

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a centered t-distribution with degree of freedom v. It should be noted that for v> 2, "" vi, the copula of Y can be written as

: v, Pv

(U) tv,

E By the theorem of Sklar

(2)

where for i, j e {1, ..., d} and where C denotes the distribution function of V ^ z / Vs, where «~ ^ and Z ~ Nd (0, p)

are independent (i.e., the conventional multivariate t distribution function) and% denotes the edge distribution function of (i.e., the conventional univariate t distribution function). In the bivariate case, the expression of the copula can be written as

2 * (i

"S ^ -2 /) i2s t + t ^

1 + - ■ r

v [1 -p \ 2

- (v + 2) / 2

dsdt. (3)

It should be noted that p12 is simply the conventional linear correlation coefficient of the corresponding bivariate distribution when v> 2. The density function of the t-copula is given by v + 1

cv, p

v + d i / dctp r

fy \ d '

V2Y

V + 1

•1

ng = i i k l + _ £

v v + d

1 +

_i \

y'p y

(4)

where> * = <; («) •

Let Hd be freely selectable continuous, strictly increasing distribution functions and let Y be given by (1) with a linear correlation matrix 2. Then

X = (fff 1 (fv (ri))

(5)

a tv-copula and marginal distributions Hi, ..., Hd. The distribution of X is called the meta-t distribution. Note that X has a t-distribution if and only if Hi, ..., Hd are univariate tv distribution functions.

The tail-dependence coefficient expresses the limiting conditional probability of the linked quantile overshoot. The t-copula has an upper and lower outlet dependency

1 2 / v-l (s / v + l ijl-Pu / - ^ / l + P] 2)>

in contrast to the Gaussian copula, which has 1 = 0. It can also be seen from the above expression that the coefficient of leakage dependence in p12 increases and decreases in v, as would be expected since a t-distribution converges to a normal distribution when v goes to infinity. Moreover, the coefficient of the upper (lower) leakage dependence goes to zero when the number of degrees of freedom goes to infinity for p12 <1.

The matching of the copulaparameters (p, v) is typically carried out as follows:

(i) Kendall's tau t (Xi; Yj) is estimated for each pair of risk factor log returns. An estimate of the parameter p in (2) is obtained from the relationship t (X,, Yj) = -arcsin ^ y) (6)

what is the validity for each distribution with strictly increasing edge distribution functions and a copula with an elliptical distribution having a density, i. essentially any meta-elliptical distribution that would be considered in an application. It should be noted that in high-dimensional applications, an estimate of (6) may need to be modified to assure positive positivity. This can be done by using the so-called eigenvalue method, i. the negative eigenvalues are replaced by a small positive number. Other matches are also possible.

(ii) Each log-return observation X ,. with its respective distribution function is transformed, e.g. Gaussian ^ (0, O |) yields, under the Meta-t assumption, a sample of a t-copula with a known p-parameter. Finally, the degree of freedom parameter v is estimated by a standard maximum likelihood estimation using (4).

In step (ii), the empirical limits or fitted distribution functions from a parametric family may be used.

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The simulation from t-copula comprises the following steps:

(i) Independently draw a random variable Z from the d-dimensional normal distribution with average zero, unit deviations and linear correlation matrix p, and a random variate U from the uniform distribution in (0,1).

(ii) Obtain R by setting «-g» By (1) we obtain a random variable Y from the t-distribution.

(iii) Finally, f \

tv (ri) .-, fv fad)

V

a random variate of the t-copula.

This meta-t assumption makes sense for risk factors of a similar type, e.g. foreign exchange rates. However, it has been found that it does not accurately describe the dependency structure for a set of risk factor log returns if the risk factors are of a very different type, for example a mix of stock market indices, foreign exchange rates and interest rates.

It is a general problem of such models that the number of available historical data is rather small, so that at least the phased dependency can hardly be modeled. Similar problems are also known from other areas, for example in the combination reinsurance portfolios, the reliability of industrial complexes or in the weather forecast.

Summary of the invention

It is therefore a technical object of the invention to provide a system and computer-implemented method for performing a risk analysis by combining a multiple of interdependent risk factors, using real historical data to match a model, using the model as a basis is used for simulations for the present or future prediction, and at least one risk measure is obtained which describes a current or future risk or a prize is obtained. The inventive method and method should be more flexible and accurate than the known systems based on the meta-t model, but without the need to use more efficient data processing equipment and without the need to have an increased number of input data based on historical data.

This is achieved by a system and a method according to claim 1, respectively 6.

The invention still uses t-copulas. In the system and method of the invention, the financial risk factors, i. the random variables, but grouped into groups of different types, and each group gets its own degree of freedom parameter. Therefore, a random vector can be obtained, which is divided into subvectors. Each subvector is properly described by a multidimensional t-distribution, where each multidimensional t-distribution has a different degree of freedom parameter, and the groups still have dependency on each other through a correlation matrix and have phased dependency. Once such a clustered t-co-pula model has been created, this model can be aligned using historical data in the same way as a t-copula model is matched, except that a maximum likelihood estimate of the plurality of degrees of freedom parameters is performed separately for each of the plurality of groups of risk factors. The simulation is then also executed in the same way as when the t-copula model is used, and the same types of values of risk masses are obtained.

It has been empirically found that the resulting values of risk masses are different from that obtained using the ordinary t-copulas when the grouped t-copulas are used. It has thus been found that the new system and method are better able to capture the risk in a large set of risk factors.

Although the present invention will be described below in conjunction with a preferred embodiment and method of use, it is to be understood that it is not intended to limit the invention to that embodiment. Instead, it is intended to cover all alternatives, modifications and equivalents which may be included within the spirit and scope of the present invention as defined in the appended claims.

Brief description of the drawings

The present invention will become more clearly understood by reference to the following detailed description of the preferred embodiments taken in conjunction with the accompanying drawings, in which: FIG

Fig. 1 illustrates the inventive system S and its input data and

Fig. 2 illustrates the inventive system S in use for multiple portfolios.

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Detailed description of the present invention

In the system and method according to the invention, z ~ Nd (0, p), where p is any linear correlation matrix, independent of U, a random variable is uniformly distributed in [0,1]. In addition, Gy denotes the distribution function of f. According to the invention, a subdivision {1, ..., d} is made in m subsets of quantities s1 sm, where m is different from 1. for k = 1, ..., m. If

Y - R \ Zsi si 2-2'si + lì "* '- ^ 2 ^ 5i + s2 s (7)

then the random vector (Y ^. ^ Ysi) 'has a Si-dimensional t-distribution with Vi degrees freedom and, for k = 1, ..., m-1, (Ys1 + ... + sk + 1, .. .Ys1 + ... + sk + 1) 'is a sk + rdimensional t-distribution with vk + i degrees of freedom. Finally, Fk denotes the distribution function of Yk and Hi, ..., Hd are any continuous distribution functions.

X = (^ TT1 (^ i (yi)), -, JPr71 (^ (^))) '

is thus a generalization of the meta-t model, which allows different subsets of the components to have different degrees of freedom parameter vm. His copula model is called subsequently grouped t-copula model.

An example of d = 4 is given, i. E. for four risk factors, where each risk factor belongs to an equity. In this example, two of these four equities are from the US, two of them are from Switzerland. This example is not representative, as these models typically comprise a multiple of risk factors.

The four risk factors are described by four random variables that are interdependent. According to the invention, the four random variables are divided into groups. Here we choose to subdivide them by country: i. we get two groups of risk factors, where each group represents one country.

To enable the execution of a simulation according to the invention, a 4d random vector Y = (Yi, Y2, Y3, Y4) with grouped t-copula dependence among the four components is needed.

By using standard techniques, the linear correlation p is determined for the four risk factors. Then, two additional parameters v1 and v2 are introduced, which take into account the run-out dependence and also allow different run-off dependencies in groups 1 and 2. For example, vi = 4, which describes a high leakage dependency, and v2 = 25, which describes a low leakage dependency.

Z = (Zi, Z2, Z3, Z4), Gì and G2 are random variables having the following distribution:

Z ~ # 4 (0, p)

in which

Xv is the usual Chi Square distribution.

U is independent of Z and uniformly distributed in [0,1]. The two random variables Ri and R2 with distribution are R1 = Gr1 (U) R2 = G2-1 (U)

Finally, form

RLZ2 R2Z3

.- ^ 2 ^ 4

This is by definition a random vector with a grouped t-copula.

The grouped t-copula can be written in a form similar to (3). However, because of the multidimensional case, the expression is rather complex and is therefore not explicitly reproduced here. The skilled person knows how to write this expression. We believe that the properties of the grouped t-copula are best understood from (7) and the above stochastic representation. Moreover, an explicit copula expression for matching the clustered t-copula model with historical data and for the simulation using the matched grouped t-copula model is not necessary, as can be seen below:

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The simulation of the grouped t-copula according to the invention is no more difficult than the simulation of a t-copula. The simulation includes the following steps:

(i) Independently draw a random variable Z from the d-dimensional normal distribution with mean zero, unit variances and linear correlation matrix p, and random variate U from the uniform distribution [0,1].

(ii) Obtain Ri, ..., Rm by setting *, - o >> for k = 1, ..., m. By (7) we obtain a random variate (Yi, ..., Yd) 'from the grouped t-distribution.

(iii) Finally, tv is 1 (iì) -; tv1 (r S, \ tv2 {y S} - 'v2 (^ 1 +' »J, Vm ^ '

a random variate from the grouped t-copula.

The matching of this model is identical to that of the meta-t distribution, except that the maximum likelihood (ML) estimation of the m degree of freedom parameters vk must be performed separately for each of the risk factor groups. The key point is that the approximation r (zi, zy) - ~ arcsm (p; y) (8)

is very accurate. Again, it may be that the eigenvalue method must be used to ensure positive positivity.

In the following, a specific example is given to clarify the process according to the invention:

We consider an internationally diversified loan portfolio with K announcements. It is believed that the systematic risk of each announcement transaction is adequately described by a set of risk factors, which are 92 country / industry equity indices, as shown in Table 1. These risk factors are divided into eight groups, which are defined by a country. The subdivision by country is only one way of group formation. Other subdivisions, such as subdivisions by industry sector, are also possible.

According to the invention, this grouped t-copula is used to describe the dependency structure of the risk factors and to complete the model by specifying normally distributed boundary values. The marginal values for monthly income are assumed to be distributed normally. Other distributions are also possible.

For our example we consider a single announcement business k and take a time horizon T = i month. Ik is the state variable for the announcement business k in the time horizon T- In this example, we only look at default events and not the impact of improvements and deteriorations on credit quality. Therefore we assume that lk takes values in {0,1}: the value 0 represents the default state, the value 1 represents the non-default state.

Yk is a random variable with continuous distribution function

Fk (x) =? [Yk ^ x \ ■

dk e R and set jk = 0 <=> Yk ^ dk (9)

The parameter dk is called the default threshold and (Yk, dk) is the latent variable model for lk. The following interpretation is applied to Yk. Let £ be the asset value of announcement business k at time t. We put j i. Yk is defined as the monthly log return of the asset value. A default occurs when the log return of the asset value falls below the threshold dk.

We find parameters ak and \ e [0,1] such that

Yk = a '^ X + ^ l - X jç sjç s] ç, (10)

where X is the vector of the monthly risk factor log return, with a grouped t-copula and normally distributed boundary values, E [X] = 0 and eK ~ N (0,1), independent of X. The model (10) says that the monthly log return from the asset value can be linked by a'k X to the risk factors, giving the systematic component of the risk and the same additional idiosyncratic component e K. The parameter Âk is the coefficient of determination for the systematic risk (how much the variance can be explained by the risk factors) and s \ = Var (yk) = a'k Cov (x) & k -

Let nk be the unconditional probability of the default of the announcement business k, i. «-« w «is assumed to be given by an internal or external rating system or other method. The conditional probability for the announcement business k can be written with given risk factor X as

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g ^ (x) = p [Yk <^ lx] = j> f Fk k),

V ^ kyft'k Cov (X) a / t ^

where <ï> denotes the standard normal cumulative distribution function. In the classical model, Yk is normally distributed, and as a result, the nk quantile f {xk) can be easily calculated. Here the distribution function of Yk is unknown: fìm is replaced by the empirical quantile estimation ^ 'M. Consequently, the estimated conditional probability of the default f'i'Xid is obtained by replacing in the equation for * «- su by.

This default model described by equations (9) and (10) is applied to each individual announcement transaction in the loan portfolio. For given risk factors X, the default message defaults are treated as conditionally independent, i. the eK's are independent.

For each Risk Factor Scenario X, the off-beat prescriptions of independent Bernouli mix distributions are simulated with parameters Of course, one could also simulate Yk using Equation (8) so that a default occurs when the simulated Yk is less than the estimated default threshold PtM. The advantage of using the Bernoulli Blend model is that it can easily be extended to a Bernoulli Blend model for a sub-portfolio of homogeneous announcements.

Lk (X) e {0,1} is the conditional default indicator for the announcement business k, Ek is the corresponding hazard and lk is the loss-given default. Then returns the total credit loss in Scenario X.

In summary, the credit loss distribution is obtained by a three-phase procedure:

(i) simulation of the monthly risk factor log returns X from a grouped t-copula with normal boundary values;

(ii) For each announcement business k, simulation of the conditional default indicator lk (X) e {0,1} from a Bernoulli mixture model with conditional default probability ê * (x);

(iii) Estimate the distribution of credit losses over a large set of scenarios for X by integrating exposures and loss-given default into loss function L (X).

In this example, we reconcile the clustered t-copula and the normally distributed marginal values using monthly risk factor log returns from 1992 to 2002 (ie, 120 observations) obtained from DataStream ™. Table 1 shows the estimated degree of freedom parameters for various subsets of risk factors and the total estimated degree of freedom parameters. Because of the difference between different subsets of degree of freedom parameters, a clustered t-copula is more suitable for describing the dependency structure.

[0056]

amount

Number of risk factors v

OFF indexes

9

15

CAN indexes

14

24

CH indices

4

19

FRA indices

5

67

GER indices

10

65

JPN indices

15

14

UK indices

15

17

US indices

20

21

Total

92

29

Table 1: Estimated degrees of freedom v for different sets of risk factors. The indices of the country equities are for large industrial sectors.

The loan portfolio contains K = 200 announcement transactions with the same unconditional default probability n = 1%. Each announcement transaction is assigned to a country, so there are 25 of each country. The weights ak and ^ (k = 1, ..., 200) are generated as follows. For XK, we randomly select values between 20% and 60%, which are common in credit modeling. Each announcement transaction is then described by two different risk factors (labeled ii and i2) from the country to which it has been assigned, and the value of 4 (and thus also of "?") Is drawn from a uniform distribution in (0,1). so that 4 + at = 1 ■ In addition, each announcement transaction has a total exposure of 1000 CHF and the loss-given default is assumed to be uniformly distributed in [0,1].

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500,000 simulations were performed using a 92-dimensional Sobol sequence. The simulated default frequencies were all in the 0.97% -1.03% range and the expected value of the loss distribution of the portfolio was estimated to be less than 0.2% error.

In the following the results are compared by means of a comparison between our new system involving the clustered t-copula (with degrees of freedom parameters as shown in Table 1) and (1) a model with t-copula (with 29 degrees of freedom) and ( 2) presented a model with Gaussian copula. The Gaussian model was taken as the baseline and the differences in the risk masses were expressed as percentages. Table 2 shows various risk measures for the overall distribution of credit losses. The consideration of the discharge dependencies with the t-copula results in an increased estimation of the risk. The introduction of the grouped t-copula even resulted in greater risk mass. The 99% failure rate is more than 10% greater than normal in this case. [0060]

measure

T29

grouped-t

Max value

29.07%

41.4%

Std. Abw.

04.02%

05.03%

95% Quanti I

01.01%

01.07%

99% Quanti I

04.03%

6.0%

95% shortage

04.03%

08.04%

99% shortage

08.09%

08.10%

Table 2: Risk mass of the sample portfolio using a t29 copula or clustered t-copula for modeling dependence among the 92 risk factors. The values shown are the percentage deviations from those obtained with the normal copula.

The invention described above is preferably performed by using a data processing system using at least one computer. This system S, as shown in Fig. 1, comprises:

Modeling and matching means Mod / Cal, which means Mod / Cal comprise a program which describes d risk factors as random variables Xi to Xd, the random variables being related by a correlation matrix p, these means

form m groups of random variables Xi to Xd,

describe the random variables Xi to Xd as d-dimensional random vector X, thereby forming m sub-vectors, each sub-vector consisting of a group of random variables Xi to Xd,

describe the dependencies of the risk factors as implicit copula of a d-dimensional random vector Y, this random vector Y consisting of m subvectors Yk (k = 1 to m, m * 1), each subvector Yk having a t-distribution with a parameter vk where this parameter describes a degree of freedom and where its copula is a t-copula, where d, m and k are natural positive numbers;

Input means (a, b, c)

for inputting or selecting match data to obtain values for the vk freedom degree parameters separately for each of the m subvectors Yk by using the modeling and matching means, and to obtain values for the correlation matrix p for all random variables Xi to Xd,

for entering or selecting at least one risk mapping function L (X), in particular a profit and loss function, and

- for entering portfolio data of the portfolio to be analyzed;

Simulation means SIM for simulating the implementation of the d risk factors by using the adjusted correlation matrix p, the adjusted values vk of the degree of freedom parameters, the risk mapping function L (X) and the portfolio data of the portfolio and

- Output means for showing the output data of the simulation in the form of a risk measure or a price.

In a preferred embodiment, the system comprises at least three input levels:

a first level comprising first input means (a) for inputting or selecting the trimming data, these data being used by the modeling and trimming means Mod / Cal;

a second level comprising second input means (b) for inputting or selecting at least one risk mapping function L (X), this function preferably being handled by risk mapping means RM; wherein said risk mapping means RM may be standalone means or part of the simulation means SIM; and

- A third level with third input means (c) for inputting the specific portfolio data, which are used by the simulation means SIM.

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The fit data is all data needed to match the model. Usually, they include historical data, boundary values and information concerning the groups to be formed, i. a maximum number of groups and information in view of the aspects and criteria of the groups are formed. The risk mapping function is preferably a profit and loss function and it depends on the general type of portfolio being analyzed. The portfolio data depends on the general type of portfolio being analyzed and can change daily.

The adjustment is performed periodically with updated data, for example once a year. The risk mapping feature only needs to be changed when a new general type of portfolio is entered. The portfolio data is entered more frequently, i. every time an updated risk measure or price is to be received. Depending on the type of business and the type of portfolio, this is usually done daily or at least once a week.

The system may be used by different users, allowing users to have different levels of mathematical understanding. The modeling and reconciliation steps are usually performed by a first person, this person usually has a fundamental mathematical background. The risk mapping step is typically performed by a second person, who is usually a well-trained, seasoned risk analyst, and preferably has some kind of mathematical background. The simulation is performed by a risk analyst responsible for the portfolio.

As can be seen in Figure 2, the system also allows to run simulations with different types of portfolios and with different portfolios within the same type, using the same modeling and depreciation means Mod / Cal. The pooling data then includes information about all the portfolios to be processed. For example, if a first portfolio comprises 50 equities from 10 countries and a second portfolio 70 equities from 20 countries, with 30 of the equities and 5 of the countries being the same as in the first portfolio, the reconciliation data considers 90 different types of equities and can have 25 groups from different countries. For each type of portfolio, there are separate risk mapping means RM for entering the specific risk mapping function L (X). For each type of portfolio, a separate simulation SIM can be performed.

In a preferred embodiment of the system S, the system comprises a data memory for storing the historical data. However, it is also possible to store the historical data on other means and to transfer the historical data to the system when the reconciliation is performed. It is also possible to use another computer or subsystem for alignment and simulation, where the data is transferred from the matching computer or matching subsystem to the simulation computer or the simulation subsystem. The system according to the invention can then comprise storage means for storing the adjusted correlation matrix p and the adjusted parameters vk, which describe the degrees of freedom. In this way simulations can be performed on different computers at the same time.

Preferably, the inventive system further comprises input means for manually grouping the d interdependent risk factors or for manually selecting a grouping from an assortment of some types of groupings. This makes it possible for a user to group risk factors by country or industry sectors or other criteria.

As can be seen, the grouping of the t-copulas allows large quantities of risk factors of different classes to be modeled. This grouped t-copula has the property that the random variables within each group have a t-copula with possibly different degrees of freedom parameters in the different groups. This results in a more flexible overall dependency structure, which is more suitable for large amounts of risk factors. The system allows the run-off dependence present in the data to be modeled more accurately than the common Gaussian copula and t-copula when it is matched for a historical risk factor record.

Claims (5)

claims 1. A data processing system for performing a risk analysis of a portfolio, the system comprising Modeling and matching means, which means are arranged to describe the risk factors as random variables up to Xd, wherein the random variables are related to each other by a correlation matrix p, and wherein these means are established - form m groups of random variables X ^ to Xd, to describe the random variables X 1 to X d as a d-dimensional random vector X, thereby forming m sub-vectors, each sub-vector consisting of a group of the random variables X 1 to X d, describe the dependencies of the risk factors as the implicit copula of a d-dimensional random vector Y, where, this random vector Y consists of m subvectors Yk and k = 1 to m, m * 1, d, m and k are positive natural numbers, each subvector Yk having a t-distribution with a parameter vk, said parameter describing a degree of freedom, and wherein its copula is a t-copula; - Input means which are set up 10 CH 696 749 A5 for inputting or selecting match data, to obtain values for the vk freedom degree parameters separately for each of the m subvectors Yk by using the modeling and matching means, and to obtain values for the correlation matrix p for all random variables X1 to Xd, for inputting or selecting at least one risk mapping function L (X), in particular a profit and loss function, and - for entering portfolio data of the portfolio to be analyzed; Simulation means arranged to simulate the implementation of the d risk factors by using the adjusted correlation matrix p, the adjusted values vk of the degree of freedom parameters, the risk mapping function L (X) and the portfolio data of the portfolio, and - Output means which are arranged to show the output data of the simulation in the form of a risk measure or a price. 2. The system of claim 1, which system comprises at least three input levels, a first level comprising first input means for inputting or selecting the match data, a second level comprising second input means for inputting or selecting the at least one risk mapping function L (X) and a third level with third input means for inputting the portfolio data. The system of any one of claims 1 or 2, which system comprises a data store for storing the historical data. The system of any of claims 1 to 3, which system further comprises input means for manually grouping the d interdependent risk factors or for manually selecting a grouping from an assortment of some types of groupings. The system of any one of claims 1 to 4, which system includes storage means for storing the matched correlation matrix p and the adjusted parameters vk describing the degrees of freedom. 11